Fast question about Circle problem
The subdivision modifier won't produce good results with triangles in the geometry. (Carlo mentioned this answer, which showcases how the modifier works on topology.)

To solve this, use quads only.




Delete the center vertice, select the edge loop and do the Grid Fill operation.

Recreate the sperical shape. Set the Cursor as Pivot (.). Place it in the imaginary center of the spherical shape. Select the flat grid geometry and press &#8679; Shift&#9095; AltS to scale to sphere.